Let X be a random variable with having a chi-squared distribution with n degrees of freedom and let y be some real number at most n. Is it known how P (X < y) behaves at least in some reasonable range of y? For instance, could one determine exactly the order of the latter probability when y=cn (for some $0\lt c \lt 1$) when n becomes large?

In the end, I am interested in the upper bound for the latter probability, but I am not sure if the usual Chernoff bound gives the correct magnitude.

Since you said that you're looking for an upper bound, it should also be noted that an examination of the proof of Cramer's theorem shows that you can actually get uniform exponential upper bounds (and not just asymptotics as stated above). In fact,
$$
P(X_n < cn) \leq e^{-n I(c)}, \quad \forall n\geq 1, \text{ and } 0 < c < 1.
$$